If f is continuous and lim x → 0 ∫ 0 x f ( t ) s i n t d t x l n ( 1 + x ) = 5 , find f ( 0 ) .

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Mathematics - Limits Question with Solution | TestHub

MathematicsLimitsStandard Limits (All types)Medium2 minQB

MathematicsMediumsingle choice

If $f$ is continuous and $lim_{x \to 0} \frac{x l n ( 1 + x )}{\int _{0}^{x} f ( t ) s i n t d t} = 5$ , find $f (0)$ .

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Answer:

Solution:

Apply LH́ospital twice, or use local equivalents. The numerator is $x^{2} + o (x^{2})$ and the denominator is $f (0) x^{2} /2 + o (x^{2})$ . Hence the limit is $2/ f (0) = 5$ , so $f (0) = 2/5$ .

Stream:JEESubject:MathematicsTopic:LimitsSubtopic:Standard Limits (All types)

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